Continuous-time matrix algorithms systolic algorithms and adaptive neural networks

In the domain of “continuous-time matrix algorithms”, matrix based algorithms are studied from the viewpoint of continuous-time systems theory and differential geometry. We put emphasis on formulas for tracking decompositions of a time-varying matrix, and present them as tools for the design and analysid of matrix algorithms. We define a class of continuous-time matrix algorithms with a uniform parallel signal flow graph. We derive algorithms for recursive least-squares estimation, belonging to this class, which are continuous-time limits of known systolic algorithms. Some of them are candidates for analog realization. For algorithms for subspace tracking, belonging to the same class, we present new analysis results based on continuous-time concepts. From these algorithms we also derive new fully pipelined systolic algorithms, inheriting the main properties of their continuous-time counterparts. We reinterpret the presented continuous-time adaptive signal processing algorithms as adaptation laws for neural networks and derive other adaptation laws, based on isospectral algorithms, for neural memory applications. Finally, we illustrate the relevance of the differential geometric approach for the design of optimization algorithms, by studying an application in robust control theory, the computation of the structured singular value.